Date: Apr 25, 2013 5:31 AM
Subject: Re: Onto [0,1]
>William Elliot wrote:
>>> More specifically, use the representation of x in C Cantor
>>> set in base 3 with only 0's and 2's in the expansion of 3,
>>> and map
>>> f: x=0.a1a2..... ---> 0.b1b2.......
>>> Wheref(bi)= 0 , if ai=0 , f(bi)=1 , if ai=2 .
>>> This f is continuous, but not absolutely continuous ( which
>>> preserves sets of measure zero. )
>>Why is f continuous?
>Because f is increasing.
Because f is _strictly_ increasing.
>If X,Y are subsets of R, and f: X -> Y is a monotonic function,
If X,Y are subsets of R, and f: X -> Y is a _strictly_ monotonic
>then f is continuous (with respect to the relative topologies on
>X and Y inherited from R).